This special issue “Multigrid Methods” in the Journal of Numerical Linear Algebra with Applications contains nine papers submitted by participants from the 19th Copper Mountain Conference on Multigrid Methods, held in the Colorado Rocky Mountains March 24–28, 2019. The papers cover a range of topics on Multigrid, including algorithm development for a variety of applications, convergence and complexity theory, and implementation on high‐performance computing platforms. The First Copper Mountain Conference on Multigrid Methods was organized by Steve McCormick in 1983. It has taken place every 2 years since then and starting in 1990 has alternated with the Copper Mountain Conference on Iterative Methods. Over the years, the two conferences have become some of the main international meetings focusing on Multigrid methods and iterative methods.

The 19th Copper Mountain Conference on Multigrid Methods took place Sunday March 24 to Thursday March 28, 2019 at the Copper Mountain Resort in Colorado, USA. The meeting organizers were Front Range Scientific Computations, Inc. and the coorganizers were the University of Colorado, the Center for Applied Scientific Computing at LLNL, and the University of Illinois at Urbana‐Champaign. The meeting's conference chairs were Robert Falgout and Luke Olson.

The first day of the meeting, Sunday, included three tutorials: (1) Multigrid—the Fundamentals given by David Moulton from Los Alamos National Lab; (2) Algebraic Multigrid and Advanced Topics given by Luke Olson from the University of Illinois at Urbana‐Champaign; and (3) Parallel Multigrid given by Rob Falgout from LLNL. The following days each had between two and four parallel sessions with a total of 105 talks. Three of the more notable talks were given by the winners of the years student paper competition: (1) Daniel Fortunato from Harvard University, (2) Peter Ohm from Tufts University, and (3) Pieterjan Robbe from Katholieke Universiteit Leuven. The other conference talks covered a range of topics, including multiscale modeling, algebraic multigrid, design, and performance of multigrid for emerging architectures, parallel time integration, coupled physics problems, structured and matrix‐free methods, hierarchical low‐rank matrix decompositions, graph problems, multilevel methods for stochastic problems, and applications to machine learning.

The special issue for this years conference consists of the nine papers.^1-9 In Reference 1, Murray and Weinzierl develop a stabilized asynchronous FAC Multigrid solver for spacetrees, that is, meshes as they are constructed from octrees and quadtrees. Paper,² by Farrell et al., concerns the local Fourier analysis of the additive Vanka type relaxation schemes that are typical choices for the discretized Stokes equations. Multilevel algorithms for computing graph embeddings are developed by Quiring and Vassilevski in the paper.³ In Reference 4, Robbe et al. present an unbiased multiindex Monte Carlo method that reuses the Multiple Semi‐Coarsened Multigrid method to find coarse‐scale solutions to resolve the high anisotropy that can arise in random fields, e.g., permeability fields. In the paper,⁵ Lee introduces a new algebraic multigrid method for solving systems of elliptic boundary‐value problems. In Reference 6, Gander and Lunet design a new parallel algorithm PARASTIELTJES (based on PARAREAL) that computes the recurrence coefficients for Gauss quadrature rules applied to arbitrary weight functions of the associated orthogonal polynomials in parallel. The paper,⁷ by Fairbanks et al., concerns the development of a multilevel Monte Carlo approach for estimating posterior moments of a particular quantity of interest, where an element agglomerated algebraic multigrid technique is used to generate the hierarchy of coarse spaces with guaranteed approximation properties for both the generation of spatially correlated random fields and the forward simulation of Darcy's law to model subsurface flow. In Reference 8, Mitchell et al. study the parallel performance of algebraic multigrid domain decomposition methods on multicore architectures. The paper,⁹ by Friedhoff and Southworth, analyzes the optimal (h‐Independent) convergence of Parareal and multigrid reduction in time using Runge–Kutta time integration.

The members of the Program Committee for the conference were James Brannick, Susanne Brenner, Marian Brezina, Craig Douglas, Van Emden Henson, Kirk Jordan, Scott MacLachlan, Tom Manteuffel, David Moulton, Kees Oosterlee, John Ruge, Ulrich Rüde, Hans De Sterck, Stefan Vandewalle, Ulrike Yang, and Irad Yavneh. The committee served as Guest Editors for the years special issue. We thank the Journal of Numerical Linear Algebra with Applications for hosting the special issue.

《数值线性代数及其应用》杂志上的特刊“ Multigrid Methods”包含2019年3月24日至28日在科罗拉多洛矶山脉举行的第19届铜山多网格方法会议的参与者提交的9篇论文。这些论文涵盖了一系列有关Multigrid的主题，包括针对各种应用程序的算法开发，收敛性和复杂性理论以及在高性能计算平台上的实现。史蒂夫·麦考密克（Steve McCormick）于1983年组织了第一届多网格方法铜山会议。此后每两年召开一次，从1990年开始与铜山迭代方法会议交替举行。多年来，这两次会议已成为针对Multigrid方法和迭代方法的一些主要国际会议。

第十九届多网格方法铜山会议于2019年3月24日至2019年3月28日星期四在美国科罗拉多州的铜山度假胜地举行。会议的组织者是Front Range Scientific Computes，Inc.，联合组织者是科罗拉多大学，LLNL应用科学计算中心和伊利诺伊大学香槟分校。会议的会议主席是Robert Falgout和Luke Olson。

会议的第一天，星期天，包括三个教程：（1）多重网格-Los Alamos国家实验室的David Moulton给出的基础知识；（2）伊利诺伊大学厄本那香槟分校的卢克·奥尔森（Luke Olson）提出的代数多重网格和高级主题；（3）LLNL的Rob Falgout提供的并行Multigrid。接下来的几天中，每两天进行两次至四个平行的会议，总共进行105次演讲。本年度学生论文竞赛的优胜者进行了三个比较著名的演讲：（1）哈佛大学的Daniel Fortunato，（2）塔夫茨大学的Peter Ohm和（3）鲁汶大学的Pieterjan Robbe。其他会议演讲涵盖了一系列主题，包括多尺度建模，代数多网格，设计以及针对新兴架构的多网格性能，并行时间集成，

今年会议的特刊由九篇论文组成。^1-9在参考文献1中，Murray和Weinzierl为空间树（即，由八叉树和四叉树构造的网格）开发了一种稳定的异步FAC多重网格求解器。Farrell等人的论文²涉及加性Vanka型松弛方案的局部傅里叶分析，这是离散Stokes方程的典型选择。本文由Quiring和Vassilevski开发了用于计算图嵌入的多级算法。³在参考文献4中，Robbe等。提出了一种无偏多指数蒙特卡罗方法，该方法重用了多半粗化多重网格方法来找到粗糙尺度的解决方案，以解决随机场（例如渗透率场）中可能出现的高各向异性。在本文中，⁵ Lee提出了一种新的代数多重网格方法，用于解决椭圆形边值问题的系统。在参考文献6中，Gander和Lunet设计了一个新的并行算法PARASTIELTJES（基于PARAREAL），该算法计算并行应用于相关正交多项式的任意加权函数的高斯正交规则的递归系数。纸⁷Fairbanks等人的论文涉及用于估计特定数量的后矩的多级蒙特卡洛方法的开发，其中使用了一种元素凝聚的代数多重网格技术来生成粗糙空间的层次结构，并为这两个生成代提供了近似的特性。空间相关随机场的建模和达西定律的正演模拟，用于模拟地下流动。在参考文献8中，Mitchell等人。研究代数多网格域分解方法在多核体系结构上的并行性能。纸张，⁹通过Friedhoff和索斯沃斯，分析最优（H-独立）Parareal的收敛性和使用龙格-库塔时间积分时间多重网格减少。

会议计划委员会的成员包括詹姆斯·布兰尼克，苏珊·布伦纳，玛丽安·布雷齐纳，克雷格·道格拉斯，范·埃姆登·汉森，柯克·乔丹，斯科特·麦克拉克兰，汤姆·曼泰菲尔，戴维·摩尔顿，基斯·奥斯特里，约翰·鲁格，乌尔里希·吕德，汉斯·德·斯特克，斯蒂芬·范德沃（Stefan Vandewalle），杨致远（Ulrike Yang）和伊拉德·雅夫妮（Irad Yavneh）。该委员会担任多年特刊的客座编辑。我们感谢带有应用程序的《数值线性代数杂志》主办这一期特刊。